Bring a large pot of salted water to a boil. Cook pasta about 1 minute short of the package instructions and drain, reserving pasta water.
Heat a large dutch oven or saute pan with 3 tablespoons of extra virgin olive oil. Add the sausage and cook, breaking up with a wooden spoon or potato masher, until browned, about 6-8 minutes.
Add the broccoli to the pan, season with salt and pepper and toss to coat in the pan drippings. Saute for 2-3 minutes, just until tender. Deglaze with a cup of reserved pasta water, scraping up brown bits from the bottom of the pan.
Add the pasta and toss to coat. Cook for 1 minute and remove from the heat.
Add the freshly grated Pecorino and a drizzle of olive oil. Toss to coat until creamy. Serve with extra Pecorino and redchili flakes.